reflection in mirror
I assume you are using Eevee as it seems in the second screenshot.
In Eevee to get better reflections you should use "Reflection Cubemaps" or "Reflection Planes".
You can find them in "add menu" > "light probe".
Since your mirror is a plane object I suggest you to put a reflection plane on the surface of the mirror. Here you can see the difference:



This is without reflection plane.
And this is with a reflection plane on the surface of the mirror object:



HDRI image should not affect objects reflections, only background reflection.

You can read abount reflection cubemap/plane here: Blender 2.81 Manual - Eevee Light Probes